Abstract
In this paper we show that if the supremal functional
is sequentially weak* lower semicontinuous on \({W^{1,\infty}(B, \mathbb{R}^d)}\) for every open set \({B \subseteq \Omega}\) (where \({\Omega}\) is a fixed open set of \({\mathbb{R}^N}\)), then \({f(x,\cdot)}\) is rank-one level convex for a.e \({x \in \Omega}\). Next, we provide an example of a weak Morrey quasiconvex function which is not strong Morrey quasiconvex. Finally we discuss the L p-approximation of a supremal functional F via \({\Gamma}\)-convergence when f is a non-negative and coercive Carathéodory function.
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Prinari, F. On the lower semicontinuity and approximation of \({L^{\infty}}\)-functionals. Nonlinear Differ. Equ. Appl. 22, 1591–1605 (2015). https://doi.org/10.1007/s00030-015-0337-y
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DOI: https://doi.org/10.1007/s00030-015-0337-y